math.GT daily digest: 5 new submissions for 9 July 2026

On 4-dimensional convex projective domains invariant by a lattice of

• Authors: Ludovic Marquis; Pierre-Louis Blayac
• arXiv: 2607.07150 2
• Subjects: Geometric Topology (math.GT); Group Theory (math.GR); Metric Geometry (math.MG)
• Comments: 28 pages, 4 figures, 4 pages Appendix. Comments welcome!

This paper is a sequel to the erratum by the authors to a paper by Crampon and Marquis (see arXiv:1202.5442). The main result of the Erratum was relating several notions of geometrical finiteness in round convex projective geometry and we prove here that our series of implications was sharp, by providing counterexamples to the implications that were not established. Our counterexamples are 4-dimensional convex domains acted on by where is a lattice of and is the irreducible representation of of dimension . We give a description of all -invariant convex domains, and in particular we construct one which is "close enough" to the convex hull of the limit set of so that the Hilbert volume of the convex core is infinite. We include an appendix with a smoothing procedure in the spirit of Cooper, Long and Tillman (arXiv:1511.06206) and Danciger, Guéritaud and Kassel (arXiv:1704.08711).

Abelianizations of finite-index subgroups of the handlebody group

• Authors: Annie Holden
• arXiv: 2607.07587 3
• Subjects: Geometric Topology (math.GT); Algebraic Topology (math.AT)
• Comments: 16 pages, 4 figures

For genus , it is an open question whether the mapping class group of a handlebody contains a finite-index subgroup with nontrivial rational abelianization. In this paper, we provide evidence that no such subgroup exists. First, we prove that, for all such finite-index subgroups , meridian multitwists vanish in . Next, we show that for finite-index subgroups containing the handlebody Torelli group, or large enough subgroups of the twist group or the handlebody Johnson kernel.

The Algebraic Montgomery-Yang Problem

• Authors: Woohyeok Jo; Jongil Park; Kyungbae Park
• arXiv: 2607.06686 4
• Subjects: Algebraic Geometry (math.AG); Geometric Topology (math.GT)
• Comments: We welcome any comments. 35 pages

We completely resolve the algebraic Montgomery - Yang problem, a conjecture of Kollar stating that every rational homology projective plane with quotient singularities and a simply-connected smooth locus has at most three singular points. The crux of our proof is a new lattice theoretic constraint, obtained by combining Donaldson's diagonalization theorem with the distinguished spin^c structure on the smooth locus whose determinant line bundle is the canonical bundle. Together with the orbifold Bogomolov - Miyaoka - Yau inequality, this constraint rules out all remaining cases in the problem and completes the proof.

Soft cells, Tubular Tilings and the Hidden Phases in Binary Mixtures

• Authors: Gábor Domokos
• arXiv: 2607.06810 5
• Subjects: Applied Physics (physics.app-ph); Geometric Topology (math.GT); Metric Geometry (math.MG)
• Comments: 25 pages, 8 figures

Biological and physical systems ranging from Fermi surfaces and skeletal structures to reaction--diffusion patterns and cosmological models may be viewed as binary mixtures in which a smooth interface separates two complementary phases. While the interface is often directly observable, the topology of one of the phases may remain hidden. To study such systems, we introduce tubular tilings, a geometric framework for discretizing binary mixtures on smooth manifolds of arbitrary dimension and topology. We prove that tubular tilings satisfy global Euler balance laws relating the topology of the ambient manifold, the discretized phases, and their interfaces. These balance laws provide a practical inference principle: topological information about a hidden phase can be recovered from the observable phase and the geometry of the separating interface. We further show that, in dimensions , tubular tilings form a subclass of soft tilings, the recently discovered class of corner-free tessellations. Applications to Fermi surfaces and cosmological shell decompositions illustrate how the theory can be used to extract otherwise inaccessible topological information about complex geometric structures.

On rank two Kleinian groups with three parabolics

• Authors: Alex Elzenaar
• arXiv: 2607.07424 6
• Subjects: Group Theory (math.GR); Geometric Topology (math.GT)
• Comments: 14 pages, 10 figures. Key words: two-bridge links, Kleinian groups, Heegaard splittings, tunnel number one links, maximal cusp groups, identification of matrix groups. Correspondence welcome

The free group of rank is the fundamental group of the genus handlebody . We study discrete representations of this group into $ \mathsf{PSL}(2,\mathbb{C}) $ so that three disjoint simple closed curves on the conformal boundary $\partial_\infty \mathcal{H} $ are sent to parabolic elements. We show that the only infinite covolume groups of this form are maximal cusp groups on the boundary of genus Schottky space. We also exhibit hitherto unexpected finite covolume groups which do not arise from Heegaard splitting presentations of tunnel number links.